Just the nonmagical damage. Taking a wound from another source does not suddenly allow previously un-regeneratable wounds to be able to be regenerated.
If you are frequently running into issues where you need to keep track of which damage can be healed by different means, I suggest using different marks on your damage track based on what sort of damage it is. For instance, using a single slash ( / ) vs an x ( X ).
So, for example, a character with regeneration takes magical damage (can't be regenerated), and marks the damage with X's to indicate that it can't be regenerated. Then later, takes a bullet wound, and marks that with just single slashes to indicate it
can be regenerated. At the end of the combat turn, after the regeneration test is made, the character can only heal up the boxes without X's.
You don't so much need to mark the individual wounds, just keep track of how much damage is from which sources, so I recommend keeping all of the damage marks such that the 'non-healable' damage is always marked as the first damage on the track. Don't change the total damage, just make sure you keep track of it in an organized manner. So, if you take 3 from a bullet, 2 from magic, then 3 from a bullet, you would have a total of 8 boxes marked, the first two would be X's and the rest slashes.
So, in the more likely scenario, with Magicians that regularly take drain (which can't be healed except through natural rest), you would see this for marking the drain damage.
Magician, already has 1 damage taken from drain.
( X )( )( )
( )( )( )
( )( )( )
( )( )
takes a bullet for 3 damage, and marks these boxes with a slash,
( X )( / )( / )
( / )( )( )
( )( )( )
( )( )
then casts a spell and takes 2 more damage from the drain. The magician would mark two more boxes as damaged, and make sure the two more boxes are marked with an "X" at the start of the damage track.
( X )( X )( X )
( / )( / )( / )
( )( )( )
( )( )
Finally, you can also do this to indicate that you've already attempted healing. After first aid and the Heal spell has been applied, mark all the boxes remaining with the hard X to indicate nothing else can be healed except from rest.